The loop group of $U(1)$, $LU(1)$ is the space of maps from the circle, $S^1$ to $U(1)$. The based loop group of $U(1)$, $\Omega U(1)$ is the space of based maps from the circle, $S^1$ to $U(1)$. It is known that for a topological space X,
\begin{equation} \pi_{n-1} (\Omega X)\cong \pi_n (X) . \end{equation}
This implies that \begin{equation} \pi_{0} (\Omega U(1))\cong\pi_1 (U(1))\cong \mathbb{Z} , \end{equation} which means that $\Omega U(1)$ is not path connected. Next, note that \begin{equation} LU(1)\cong\Omega U(1) \times U(1) \end{equation} (as spaces, not topological groups; note that this relation holds for loop groups of topological groups, but not necessarily for other loop spaces in general). Then \begin{equation} \pi_{0} (L U(1))\cong\pi_{0} (\Omega U(1)) \times \pi_{0}(U(1))\cong \mathbb{Z} \end{equation} which means the loop group of $U(1)$ is not path connected either.
Let's write the elements of $U(1)$ as \begin{equation} e^{i\lambda}. \end{equation} I can then write the elements of $L U(1)$ explicitly as \begin{equation} e^{i\lambda(t)}, \end{equation} where $t=e^{i\theta}$ parametrizes the circle $S^1$. I can then Fourier expand to obtain any element of $\Omega U(1)$ in the form \begin{equation} e^{i\sum_n \lambda_{n}e^{in\theta}}. \end{equation} However, if I set all the $\lambda_n$ to zero, I obtain the identity of $LU(1)$. This seems to me to contradict the fact that $LU(1)$ is not path connected, since all elements of $LU(1)$ are continuously connected to its identity element. Could someone explain?
$\lambda(t)$ does not necessarily have a Fourier series because it is not necessarily a periodic function. The condition on $\lambda(t)$ that makes $e^{i \lambda(t)}$ represent a function $S^1 \to U(1)$ is not that $\lambda(t + 2\pi) = \lambda(t)$ but that
$$\lambda(t + 2\pi) = \lambda(t) + 2 \pi k$$
for some integer $k$ (the winding number of $\lambda$). The different values of $k$ label the different path components of the loop group, so when you implicitly assume that $k = 0$ you only end up in one component.